Simplest Cubic Number Fields

TL;DR

This paper investigates which integers can serve as norms of principal ideals in specific cubic fields, simplifying the process of constructing certain unramified quadratic extensions.

Contribution

It identifies integers that do not occur as norms in these cubic fields, advancing understanding of their ideal class structure.

Findings

Certain integers are proven not to be norms in these cubic fields.

Results facilitate the construction of unramified quadratic extensions.

Simplifies previous methods for analyzing these fields.

Abstract

In this paper we intend to show that certain integers do not occur as the norms of principal ideals in a family of cubic fields studied by Cohn, Shanks, and Ennola. These results will simplify the construction of certain unramified quadratic extensions of such fields.